Helicity-supported stationary spacetimes: A class of finite-energy, horizonless, axisymmetric solutions

Abstract

We construct a class of stationary, axisymmetric, horizonless spacetimes whose curvature is generated entirely by smooth, localised differential rotation $\Omega(r)$, while the spatial geometry remains exactly flat. Despite vanishing ADM mass, these helicity-supported configurations exhibit non-trivial curvature, finite tidal forces, and a gravitomagnetic field arising from the radial shear of the rotation. The twisted stationary Killing congruence produces global frame-dragging, including a gravitational Sagnac effect, and the effective potential admits stable circular orbits for null and timelike particles. The tidal tensor gives oscillatory restoring forces, ensuring stability against radial perturbations. Linearising the Einstein equations yields a wave equation for axisymmetric perturbations of $\Omega(r)$; the effective potential is positive and localised, the operator is self-adjoint and positive definite, and the frequency spectrum is real, implying linear stability. Perturbations propagate as shear waves analogous to Alfv\'en waves. These results show that differential rotation alone can sustain a regular, asymptotically flat gravitational field with rich dynamics. This class of spacetimes provides a tractable platform for exploring gravitomagnetism, tidal and wave phenomena in smooth rotating backgrounds, with direct applications to rotating astrophysical structures.